Finiteness properties and relatively hyperbolic groups

We show that properties F n $F_n$ and F P n $FP_n$ hold for a relatively hyperbolic group if and only if they hold for all the peripheral subgroups. As an application we show that there are at least countably many distinct quasi‐isometry classes of one‐ended non‐amenable groups that are type F n $F_n$ but not F n + 1 $F_{n+1}$ and similarly of type F P n $FP_n$ and not F P n + 1 $FP_{n+1}$ for all positive integers n $n$ .